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\subsection{The special case of Newton's linearization of~(\ref{eq:toto1}) with FirstOrderType2R~(\ref{first-DS2})}
Let us now proceed with the time discretization of~(\ref{eq:toto1}) with FirstOrderType2R~(\ref{first-DS2}) by a fully implicit scheme :
\begin{equation}
\begin{array}{l}
\label{eq:mlcp2-toto1-DS2}
M x_{k+1} = M x_{k} +h\theta f(x_{k+1},t_{k+1})+h(1-\theta) f(x_k,t_k) + h \gamma r(t_{k+1})
+ h(1-\gamma)r(t_k) \\[2mm]
y_{k+1} = h(t_{k+1},x_{k+1},\lambda _{k+1}) \\[2mm]
r_{k+1} = g(t_{k+1},\lambda_{k+1})\\[2mm]
\end{array}
\end{equation}
\paragraph{Newton's linearization of the first line of~(\ref{eq:mlcp2-toto1-DS2})} The linearization of the first line of the problem~(\ref{eq:mlcp2-toto1-DS2}) is similar to the previous case so that (\ref{eq:rfree-2}) is still valid.
\paragraph{Newton's linearization of the second line of~(\ref{eq:mlcp2-toto1-DS2})} The linearization of the second line of the problem~(\ref{eq:mlcp2-toto1-DS2}) is similar to the previous case so that (\ref{eq:NL11y}) is still valid.
\paragraph{Newton's linearization of the third line of~(\ref{eq:mlcp2-toto1-DS2})}
Since $ K^{\alpha}_{k+1} = \nabla_xg(t_{k+1},\lambda ^{\alpha}_{k+1}) = 0 $, the linearization of the third line of (\ref{eq:mlcp2-toto1-DS2}) reads as
\begin{equation}
\label{eq:mlcp2-rrL}
\begin{array}{l}
\boxed{r^{\alpha+1}_{k+1} = g(t_{k+1},\lambda ^{\alpha}_{k+1}) + B^{\alpha}_{k+1} ( \lambda^{\alpha+1}- \lambda^{\alpha}_{k+1} )}
\end{array}
\end{equation}
\paragraph{Reduction to a linear relation between $x^{\alpha+1}_{k+1}$ and
$\lambda^{\alpha+1}_{k+1}$}
Inserting (\ref{eq:mlcp2-rrL}) into~(\ref{eq:rfree-11}), we get the following linear relation between $x^{\alpha+1}_{k+1}$ and
$\lambda^{\alpha+1}_{k+1}$, we get the linear relation
\begin{equation}
\label{eq:mlcp2-rfree-13}
\begin{array}{l}
\boxed{ x^{\alpha+1}_{k+1}\stackrel{\Delta}{=} x^\alpha_p + \left[ h \gamma (W^{\alpha}_{k+1})^{-1} B^{\alpha}_{k+1} \lambda^{\alpha+1}_{k+1}\right]}
\end{array}
\end{equation}
with
\begin{equation}
\boxed{x^\alpha_p \stackrel{\Delta}{=} h\gamma(W^{\alpha}_{k+1} )^{-1}\left[g(t_{k+1},\lambda^{\alpha}_{k+1})
-B^{\alpha}_{k+1} (\lambda^{\alpha}_{k+1}) \right ] +x^\alpha_{\free}}
\end{equation}
and
\begin{equation}
\label{eq:mlcp2-NL9}
\begin{array}{l}
W^{\alpha}_{k+1} \stackrel{\Delta}{=} M-h\theta A^{\alpha}_{k+1}\\
\end{array}
\end{equation}
\paragraph{Reduction to a linear relation between $y^{\alpha+1}_{k+1}$ and
$\lambda^{\alpha+1}_{k+1}$}
Inserting (\ref{eq:mlcp2-rfree-13}) into (\ref{eq:NL11y}), we get the following linear relation between $y^{\alpha+1}_{k+1}$ and $\lambda^{\alpha+1}_{k+1}$,
\begin{equation}
\begin{array}{l}
y^{\alpha+1}_{k+1} = y_p + \left[ h \gamma C^{\alpha}_{k+1} ( W^{\alpha}_{k+1})^{-1} B^{\alpha}_{k+1} + D^{\alpha}_{k+1} \right]\lambda^{\alpha+1}_{k+1}
\end{array}
\end{equation}
with
\begin{equation}\boxed{
y_p = y^{\alpha}_{k+1} -\mathcal R^{\alpha}_{yk+1} + C^{\alpha}_{k+1}(x_q) -
D^{\alpha}_{k+1} \lambda^{\alpha}_{k+1} }
\end{equation}
\textcolor{red}{
\begin{equation}
\boxed{ x^\alpha_q= x^\alpha_p - x^{\alpha}_{k+1}\label{eq:mlcp2-xqq}}
\end{equation}
}
\subsection{The special case of Newton's linearization of~(\ref{eq:toto1}) with FirstOrderType1R~(\ref{first-DS1})}
Let us now proceed with the time discretization of~(\ref{eq:toto1}) with FirstOrderType1R~(\ref{first-DS1}) by a fully implicit scheme :
\begin{equation}
\begin{array}{l}
\label{eq:mlcp3-toto1-DS1}
M x_{k+1} = M x_{k} +h\theta f(x_{k+1},t_{k+1})+h(1-\theta) f(x_k,t_k) + h \gamma r(t_{k+1})
+ h(1-\gamma)r(t_k) \\[2mm]
y_{k+1} = h(t_{k+1},x_{k+1}) \\[2mm]
r_{k+1} = g(t_{k+1}\lambda_{k+1})\\[2mm]
\end{array}
\end{equation}
The previous derivation is valid with $ D^{\alpha}_{k+1} =0$.
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